Basic Math Examples

$4 \sqrt{72} + 4 \sqrt{128 m^{2}} - \sqrt{96 n^{4} m^{3}} + 4 \sqrt{8}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 1.1.1

Factor $36$ out of $72$ .

$4 \sqrt{36 (2)} + 4 \sqrt{128 m^{2}} - \sqrt{96 n^{4} m^{3}} + 4 \sqrt{8}$

Step 1.1.2

Rewrite $36$ as $6^{2}$ .

$4 \sqrt{6^{2} \cdot 2} + 4 \sqrt{128 m^{2}} - \sqrt{96 n^{4} m^{3}} + 4 \sqrt{8}$

Step 1.2

Pull terms out from under the radical.

$4 (6 \sqrt{2}) + 4 \sqrt{128 m^{2}} - \sqrt{96 n^{4} m^{3}} + 4 \sqrt{8}$

Step 1.3

Multiply $6$ by $4$ .

$24 \sqrt{2} + 4 \sqrt{128 m^{2}} - \sqrt{96 n^{4} m^{3}} + 4 \sqrt{8}$

Step 1.4

Rewrite $128 m^{2}$ as ${(8 m)}^{2} \cdot 2$ .

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Step 1.4.1

Factor $64$ out of $128$ .

$24 \sqrt{2} + 4 \sqrt{64 (2) m^{2}} - \sqrt{96 n^{4} m^{3}} + 4 \sqrt{8}$

Step 1.4.2

Rewrite $64$ as $8^{2}$ .

$24 \sqrt{2} + 4 \sqrt{8^{2} \cdot 2 m^{2}} - \sqrt{96 n^{4} m^{3}} + 4 \sqrt{8}$

Step 1.4.3

Move $2$ .

$24 \sqrt{2} + 4 \sqrt{8^{2} m^{2} \cdot 2} - \sqrt{96 n^{4} m^{3}} + 4 \sqrt{8}$

Step 1.4.4

Rewrite $8^{2} m^{2}$ as ${(8 m)}^{2}$ .

$24 \sqrt{2} + 4 \sqrt{{(8 m)}^{2} \cdot 2} - \sqrt{96 n^{4} m^{3}} + 4 \sqrt{8}$

Step 1.5

Pull terms out from under the radical.

$24 \sqrt{2} + 4 (8 m \sqrt{2}) - \sqrt{96 n^{4} m^{3}} + 4 \sqrt{8}$

Step 1.6

Multiply $8$ by $4$ .

$24 \sqrt{2} + 32 (m \sqrt{2}) - \sqrt{96 n^{4} m^{3}} + 4 \sqrt{8}$

Step 1.7

Rewrite $96 n^{4} m^{3}$ as ${(4 n^{2} m)}^{2} \cdot (6 m)$ .

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Step 1.7.1

Factor $16$ out of $96$ .

$24 \sqrt{2} + 32 m \sqrt{2} - \sqrt{16 (6) n^{4} m^{3}} + 4 \sqrt{8}$

Step 1.7.2

Rewrite $16$ as $4^{2}$ .

$24 \sqrt{2} + 32 m \sqrt{2} - \sqrt{4^{2} \cdot 6 n^{4} m^{3}} + 4 \sqrt{8}$

Step 1.7.3

Rewrite $n^{4}$ as ${(n^{2})}^{2}$ .

$24 \sqrt{2} + 32 m \sqrt{2} - \sqrt{4^{2} \cdot 6 {(n^{2})}^{2} m^{3}} + 4 \sqrt{8}$

Step 1.7.4

Factor out $m^{2}$ .

$24 \sqrt{2} + 32 m \sqrt{2} - \sqrt{4^{2} \cdot 6 {(n^{2})}^{2} (m^{2} m)} + 4 \sqrt{8}$

Step 1.7.5

Move $6$ .

$24 \sqrt{2} + 32 m \sqrt{2} - \sqrt{4^{2} {(n^{2})}^{2} m^{2} \cdot 6 m} + 4 \sqrt{8}$

Step 1.7.6

Rewrite $4^{2} {(n^{2})}^{2} m^{2}$ as ${(4 n^{2} m)}^{2}$ .

$24 \sqrt{2} + 32 m \sqrt{2} - \sqrt{{(4 n^{2} m)}^{2} \cdot 6 m} + 4 \sqrt{8}$

Step 1.7.7

Add parentheses.

$24 \sqrt{2} + 32 m \sqrt{2} - \sqrt{{(4 n^{2} m)}^{2} \cdot (6 m)} + 4 \sqrt{8}$

Step 1.8

Pull terms out from under the radical.

$24 \sqrt{2} + 32 m \sqrt{2} - (4 n^{2} m \sqrt{6 m}) + 4 \sqrt{8}$

Step 1.9

Multiply $4$ by $- 1$ .

$24 \sqrt{2} + 32 m \sqrt{2} - 4 (n^{2} m \sqrt{6 m}) + 4 \sqrt{8}$

Step 1.10

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.10.1

Factor $4$ out of $8$ .

$24 \sqrt{2} + 32 m \sqrt{2} - 4 n^{2} m \sqrt{6 m} + 4 \sqrt{4 (2)}$

Step 1.10.2

Rewrite $4$ as $2^{2}$ .

$24 \sqrt{2} + 32 m \sqrt{2} - 4 n^{2} m \sqrt{6 m} + 4 \sqrt{2^{2} \cdot 2}$

Step 1.11

Pull terms out from under the radical.

$24 \sqrt{2} + 32 m \sqrt{2} - 4 n^{2} m \sqrt{6 m} + 4 (2 \sqrt{2})$

Step 1.12

Multiply $2$ by $4$ .

$24 \sqrt{2} + 32 m \sqrt{2} - 4 n^{2} m \sqrt{6 m} + 8 \sqrt{2}$

Step 2

Add $24 \sqrt{2}$ and $8 \sqrt{2}$ .

$32 m \sqrt{2} - 4 n^{2} m \sqrt{6 m} + 32 \sqrt{2}$